Eigenvalue inequalities for differences of means of Hilbert space operators (Q665970)

Given bounded linear operators \(A,B\) on a Hilbert space such that \(A\geq B>O \) and \(A-B\) is compact, and a scalar \(\mu\in (0,1)\), the authors consider the \(\mu\)-weighted arithmetic (resp., geometric) mean \(A\nabla _{\mu} B = (1-\mu)A+\mu B\) (resp., \(A\sharp _{\mu}B=A^{1/2}(A^{-1/2}BA^{-1/2})^{\mu}A^{1/2}\)) and the \(\mu\)-weighted Heinz mean \(H_{\mu}(A,B)=1/2(A\sharp _{\mu}B +A\sharp _{1-\mu}B)\). The estimates for the singular values of the differences \(A\nabla _{\mu} B -A\sharp _{\mu}B\), \((A\nabla _{\mu} B)A^{-1}(A \nabla _{\mu} B) -A\sharp _{2 \mu}B \), \(A\nabla _{\mu}(BA^{-1}B)-A\sharp _{2 \mu}B \), \(B\nabla _{\mu}(AB^{-1}A)-B\sharp _{2 \mu}A\) and \(\frac{A+B}{2} - H_{\mu}(A,B)\) are expressed in terms of the singular values of the operators \(A^{-1/2}(A-B)^{2}A^{-1/2}\) or \(B^{-1/2}(A-B)^{2}B^{-1/2}\). Equality conditions for the corresponding inequalities are also obtained.